We prove spectral, stochastic and mean curvature estimates for complete m-submanifolds ϕ : M → N of n-manifolds with a pole N in terms of the comparison isoperimetric ratio I_m and the extrinsic radius r_ϕ ≤ ∞. Our proof holds for the bounded case r_ϕ < ∞, recovering the known results, as well as for the unbounded case r ϕ = ∞. In both cases, the fundamental ingredient in these estimates is the integrability over (0,r ϕ ) of the inverse I^{−1}_m of the comparison isoperimetric radius. When r_ϕ = ∞, this condition is guaranteed if N is highly negatively curved.
Pacelli Bessa, G., Pigola, S., Setti, A. (2014). On submanifolds of highly negatively curved spaces. INTERNATIONAL JOURNAL OF MATHEMATICS, 25(6), 1-15 [10.1142/S0129167X14500554].
On submanifolds of highly negatively curved spaces
Pigola S.;
2014
Abstract
We prove spectral, stochastic and mean curvature estimates for complete m-submanifolds ϕ : M → N of n-manifolds with a pole N in terms of the comparison isoperimetric ratio I_m and the extrinsic radius r_ϕ ≤ ∞. Our proof holds for the bounded case r_ϕ < ∞, recovering the known results, as well as for the unbounded case r ϕ = ∞. In both cases, the fundamental ingredient in these estimates is the integrability over (0,r ϕ ) of the inverse I^{−1}_m of the comparison isoperimetric radius. When r_ϕ = ∞, this condition is guaranteed if N is highly negatively curved.
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